Abstract
A cycle cover of a directed graph is a collection of node disjoint cycles such that every node is part of exactly one cycle. A k-cycle cover is a cycle cover in which every cycle has length at least k. While deciding whether a directed graph has a 2-cycle cover is solvable in polynomial time, deciding whether it has a 3-cycle cover is already NP-complete. Given a directed graph with nonnegative edge weights, a maximum weight 2-cycle cover can be computed in polynomial time, too. We call the corresponding optimization problem of finding a maximum weight 3-cycle cover Max-3-DCC.
In this paper we present two polynomial time approximation algorithms for Max-3-DCC. The heavier of the 3-cycle covers computed by these algorithms has at least a fraction of \( \frac{3} {5} - \in \) for any ∈> 0, of the weight of a maximum weight 3-cycle cover.
As a lower bound, we prove that Max-3-DCC is APX-complete, even if the weights fulfil the triangle inequality.
Birth name: Bodo Siebert. Supported by DFG research grant Re 672/3.
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Bläser, M., Manthey, B. (2002). Two Approximation Algorithms for 3-Cycle Covers. In: Jansen, K., Leonardi, S., Vazirani, V. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2002. Lecture Notes in Computer Science, vol 2462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45753-4_6
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DOI: https://doi.org/10.1007/3-540-45753-4_6
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